Modern Analysis, Homework 3
1. Recall that a function
f
on an interval [
a, b
] is of
bounded variation
if
sup
n
∈
N
x
0
=
a<x
1
<
···
<x
n
=
b
n
X
i
=0

f
(
x
i
)

f
(
x
i
+1
)

<
+
∞
.
In such a case, we write
f
∈
BV[
a, b
]. Given
f
∈
BV[
a, b
], for
x
∈
[
a, b
], define
T
f
(
x
) =
sup
n
∈
N
x
0
=
a<x
1
<
···
<x
n
=
x
n
X
i
=0

f
(
x
i
)

f
(
x
i
+1
)

(the
total variation of
f
on
[
a, x
]).
(a) Prove that
T
f
(
x
) +
f
(
x
) and
T
f
(
x
)

f
(
x
) are increasing functions on [
a, b
].
(b) Prove that
f
∈
BV[
a, b
] if and only if
f
can be written as a difference of increasing
functions.
2. Let
{
f
n
}
n
∈
N
be a sequence of continuous functions defined on a compact metric space
(
K, d
). Suppose that for every convergent sequence of points
{
x
n
}
n
∈
N
such that
x
n
→
x
,
we have
lim
n
→∞
f
n
(
x
n
) =
f
(
x
)
where
f
is some continuous function on
K
. Prove that
f
n
→
f
uniformly on
K
.
3. Let
g
be a continuous function on [0
,
1] with
g
(1) = 0. Prove that the sequence
{
g
(
x
)
x
n
}
converges uniformly on [0
,
1].
4.
(a) Suppose that
f
is a 2
π
periodic function on
R
, Riemannintegrable on [

π, π
].
Let
ˆ
f
(
n
) denote the
n
th Fourier coefficient of
f
(ie
ˆ
f
(
n
) =
1
2
π
R
π

π
f
(
x
)
e

inx
dx
),
and suppose that
ˆ
f
(
n
) = 0 for all
n
∈
Z
.
Prove that
f
(
x
0
) = 0 whenever
f
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 Summer '11
 Staff
 Calculus, Continuous function, Wolfram Alpha, nth Fourier coeﬃcient

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